| Foreword |
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v | |
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Professor Ross C. McPhedran |
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| Foreword |
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vii | |
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| Preface |
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xvii | |
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1 | (26) |
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Conventional Optical Fibres |
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1 | (8) |
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1 | (2) |
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3 | (1) |
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3 | (1) |
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3 | (1) |
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4 | (1) |
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5 | (3) |
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8 | (1) |
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9 | (3) |
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One dimension: Bragg mirrors |
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10 | (1) |
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Photonic crystals in two and three dimensions |
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11 | (1) |
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Guiding Light in a Fibre with Photonic Crystals |
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12 | (4) |
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13 | (1) |
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Photonic crystal fibres and hollow core microstructured optical fibres |
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13 | (1) |
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14 | (2) |
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16 | (3) |
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16 | (1) |
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Main properties and applications |
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17 | (2) |
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19 | (8) |
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19 | (1) |
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Modes of a leaky structure |
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20 | (1) |
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Heuristic approach to physical properties of leaky modes |
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21 | (2) |
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Mathematical considerations |
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23 | (2) |
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25 | (2) |
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Electromagnetism -- Prerequisites |
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27 | (62) |
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27 | (7) |
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Maxwell equations in vacuo |
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27 | (2) |
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Maxwell equations in idealized matter |
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29 | (1) |
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Mesoscopic homogenization |
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29 | (1) |
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Dispersion relations -- Kramers-Kronig relations |
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30 | (4) |
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The Monodimensional Case (Modes, Dispersion Curves) |
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34 | (21) |
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34 | (1) |
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A special feature of the 1D-case: the decoupling of modes |
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34 | (2) |
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Physics and functional spaces |
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36 | (2) |
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38 | (1) |
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39 | (1) |
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Localisation of constants of propagation |
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39 | (2) |
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How can one practically get the dispersion curves and the modes? |
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41 | (1) |
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41 | (6) |
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47 | (1) |
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Modes in a binary multilayered structure |
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48 | (1) |
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49 | (1) |
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49 | (1) |
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49 | (4) |
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The variational formulation (weak formulation) |
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53 | (2) |
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An example of a Hilbert-basis in L2(R): the Hermite polynomials |
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55 | (1) |
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The Two-Dimensional Vectorial Case (general case) |
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55 | (8) |
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Some useful relations between transverse and axial components |
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57 | (3) |
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Equations of propagation involving only the axial components |
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60 | (1) |
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What are the special features of isotropic microstructured fibers? |
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61 | (2) |
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The Two-Dimensional Scalar Case (weak guidance) |
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63 | (1) |
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64 | (8) |
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64 | (1) |
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65 | (1) |
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66 | (1) |
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67 | (5) |
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Discrete finite element formulation |
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72 | (1) |
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72 | (17) |
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The crystalline structure |
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72 | (1) |
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Waves in a homogeneous space |
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73 | (1) |
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Bloch modes of a photonic crystal |
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74 | (4) |
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Computation of the band structure |
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78 | (3) |
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A simple 1D illustrative example: the Kronig-Penney model |
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81 | (8) |
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89 | (68) |
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Finite Elements: Basic Principles |
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89 | (18) |
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A one-dimensional naive introduction |
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90 | (3) |
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Multi-dimensional scalar elliptic problems |
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93 | (1) |
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Weak formulation of problems involving a Laplacian |
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93 | (1) |
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94 | (2) |
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The finite element method |
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96 | (2) |
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98 | (1) |
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99 | (2) |
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101 | (6) |
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The Geometric Structure of Electromagnetism and Its Discrete Analog |
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107 | (15) |
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108 | (1) |
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109 | (1) |
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110 | (4) |
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114 | (3) |
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Differential complexes: from de Rham to Whitney |
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117 | (5) |
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122 | (13) |
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Building the matrices (discrete Hodge operator and material properties) |
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122 | (2) |
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124 | (1) |
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125 | (5) |
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Nedelec edge elements vs. Whitney 1-forms |
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130 | (2) |
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132 | (1) |
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Transformation method for infinite domains |
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133 | (1) |
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Perfectly Matched Layer (PML) |
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134 | (1) |
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Propagation Modes Problems in Dielectric Waveguides |
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135 | (11) |
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Weak and discrete electric field formulation |
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136 | (4) |
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140 | (3) |
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143 | (1) |
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Looking for β with k0 given |
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143 | (1) |
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Discrete magnetic field formulation |
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144 | (1) |
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Eliminating one component with the divergence |
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144 | (2) |
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146 | (1) |
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146 | (7) |
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146 | (2) |
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148 | (2) |
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150 | (1) |
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Direct determination of the periodic part |
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151 | (2) |
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153 | (3) |
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156 | (1) |
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157 | (48) |
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157 | (1) |
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The Multipole Formulation |
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158 | (13) |
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The geometry of the modelled microstructured optical fiber |
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158 | (2) |
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The choice of the propagating electromagnetic fields |
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160 | (1) |
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A simplified approach of the Multipole Method |
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160 | (1) |
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160 | (2) |
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Physical interpretation of Fourier-Bessel series (no inclusion) |
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162 | (1) |
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163 | (1) |
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Fourier-Bessel series and one inclusion: scattering operator |
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164 | (1) |
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Fourier-Bessel series and two inclusions: the Multipole Method |
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164 | (2) |
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Rigourous formulation of the field identities |
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166 | (3) |
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Boundary conditions and field coupling |
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169 | (1) |
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Derivation of the Rayleigh Identity |
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170 | (1) |
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Symmetry Properties of MOF |
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171 | (4) |
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Symmetry properties of modes |
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171 | (4) |
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175 | (6) |
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176 | (1) |
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Dispersion characteristics |
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177 | (1) |
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Using the symmetries within the Multipole Method |
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178 | (1) |
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Another way to obtain sm(β) |
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179 | (1) |
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Software and computational demands |
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180 | (1) |
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Validation of the Multipole Method |
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181 | (3) |
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Convergence and self-consistency |
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181 | (3) |
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Comparison with other methods |
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184 | (1) |
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184 | (14) |
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A detailed C6v example: the six hole MOF |
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184 | (11) |
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A C2v example: a birefringent MOF |
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195 | (2) |
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A C4v example: a square MOF |
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197 | (1) |
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Six Hole Plain Core MOF Example: Supercell Point of View |
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198 | (3) |
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201 | (4) |
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205 | (20) |
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Genesis of Baron Strutt's Algorithm |
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205 | (1) |
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Common Features Shared by Multipole and Rayleigh Methods |
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206 | (3) |
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Specificity of Lord Rayleigh's Algorithm |
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209 | (1) |
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Green's Function Associated with a Periodic Lattice |
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210 | (1) |
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Some Absolutely Convergent Lattice Sums |
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211 | (2) |
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213 | (2) |
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215 | (1) |
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Normalisation of the Rayleigh System |
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216 | (1) |
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Convergence of the Multipole Method |
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217 | (2) |
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Higher-order Approximations, Photonic Band Gaps for Out-of-plane Propagation |
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219 | (1) |
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Conclusion and Perspectives |
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220 | (5) |
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A la Cauchy Path to Pole Finding |
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225 | (22) |
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A Simple Extension: Poles of Matrices |
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228 | (8) |
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232 | (1) |
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Multiple poles inside the loop |
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233 | (1) |
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Miracles sometimes happen |
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234 | (2) |
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Cauchy integrals for operators |
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236 | (2) |
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238 | (6) |
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244 | (3) |
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Basic Properties of Microstructured Optical Fibres |
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247 | (34) |
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Basic Properties of the Losses |
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247 | (3) |
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Single-Modedness of Solid Core C6v MOF |
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250 | (4) |
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A cutoff for the second mode |
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251 | (2) |
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A phase diagram for the second mode |
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253 | (1) |
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Modal Cutoff of the Fundamental Mode |
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254 | (9) |
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Existence of a new kind of cutoff |
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254 | (5) |
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A phase diagram for the fundamental mode |
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259 | (2) |
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Simple physical models below and above the transition region |
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261 | (2) |
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263 | (8) |
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Material and waveguide chromatic dispersion |
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264 | (3) |
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The influence of the number of rings Nr on chromatic dispersion |
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267 | (1) |
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A more accurate MOF design procedure |
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268 | (3) |
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A Hollow Core MOF with an Air-Guided Mode |
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271 | (6) |
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The photonic crystal cladding |
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271 | (1) |
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272 | (5) |
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277 | (4) |
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281 | (2) |
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Appendix A A Formal Framework for Mixed Finite Element Methods |
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283 | (4) |
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Appendix B Some Details of the Multipole Method Derivation |
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287 | (8) |
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Derivation of the Wijngaard Identity |
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287 | (2) |
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289 | (1) |
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Cylinder to cylinder conversion |
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289 | (1) |
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Jacket to cylinder conversion |
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289 | (1) |
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Cylinder to jacket conversion |
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290 | (1) |
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Boundary Conditions: Reflection Matrices |
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290 | (5) |
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Appendix C A Pot-Pourri of Mathematics |
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295 | (28) |
| Bibliography |
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323 | (14) |
| Index |
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337 | |
